Express $z_1=-8\sqrt{3}+8i$ in polar form. Express your answer in exact terms, using degrees, where your angle is between $0^\circ$ and $360^\circ$, inclusive. $z_1=$
Answer: The Strategy A complex number in rectangular form, $z={a}+{b}i$, can be written in polar form as $z={r}[\cos{\theta}+i\sin{\theta}]$, where ${r}$ is the absolute value, or modulus, and ${\theta}$ is the angle, or argument. Therefore, ${r}$ and ${\theta}$ can be found using the following formulas: ${r}=\sqrt{{a}^2+{b}^2}$ $\tan{\theta}=\dfrac{{b}}{{a}}$ [How did we get these equations?] Similarly, a complex number in polar form, $z={r}[\cos{\theta}+i\sin{\theta}]$, can be written in rectangular form as $z={a}+{b}i$, using the following formulas: ${a}={r}\cos{\theta}$ ${b}={r}\sin{\theta}$ [How did we get these equations?] Finding $r$ For $z_1={-8\sqrt{3}}+{8}i$ : ${a} = {-8\sqrt{3}}$ ${b} = {8}$ Therefore, we can find ${r}$ as follows. $\begin{aligned}{r}&=\sqrt{{a}^2+{b}^2} \\\\&=\sqrt{({-8\sqrt{3}})^2+({8})^2} \\\\&=\sqrt{192+64} \\\\&={\sqrt{256}} \\\\&={16}\end{aligned}$ Finding $\theta$ Using the formula, we have: $\begin{aligned}{\theta}&=\arctan\left(\dfrac{{b}}{{a}}\right) \\\\&=\arctan\left(\dfrac{{8}}{{-8\sqrt{3}}}\right) \\\\&={-30^\circ}\end{aligned}$ Since ${a}$ is negative and ${b}$ is positive, ${\theta}$ must lie in Quadrant $\text{II}$. Therefore its angle must be between $90^\circ$ and $180^\circ$. Using the identity $\tan(180+\theta)=\tan(\theta)$, we know that the following is also a solution of the equation. $180^\circ+(-30^\circ)=150^\circ$ So $\theta = 150^{\circ}$. Summary $z_1={16}[\cos{150^\circ}+i\sin{150^\circ}]$